Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

Abstract

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\mathbb{U}$, namely Theorems A and G: Theorem A (Approximation): The group of isometry $\mathit{ISO}(\mathbb{U})$ contains everywhere dense locally finite subgroup; Theorem G (Globalization): For each finite metric space F there exists another finite metric space $\overline{F}$ and isometric imbedding j of F to $\overline{F}$ such that isometry j induces the imbedding of the group monomorphism of the group of isometries of the space F to the group of isometries of space $\overline{F}$ and each partial isometry of F can be extended up to global isometry in $\overline{F}$. The fact that Theorem G, is true was announced in 2005 by author without proof, and was proved by S. Solecki in [S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005) 315–332] (see also [V. Pestov, The isometry group of the Urysohn space as a Lévy group, Topology Appl. 154 (10) (2007) 2173–2184; V. Pestov, A theorem of Hrushevski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space, math/0702207]) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from [P. Cameron, A. Vershik, Some isometry groups of Urysohn spaces, Ann. Pure Appl. Logic 143 (1–3) (2006) 70–78].